14 - Perfect and Tilting DG Modules over NC DG Rings

Summary

We begin with a study of algebraically perfect DG modules over a NC (noncommutative) DG ring A. There is a theorem giving several conditions on a DG A-module L that are equivalent to being algebraically perfect; one of them is that L is a compact object of D(A). When A is a ring, we prove that L is algebraically perfect iff it is isomorphic, in D(A), to a bounded complex of finitely generated projective A-modules. In Section 14.2 we prove a general Derived Morita Theorem.From Section 14.3 to the end of this chapter we assume that the DG rings in question are K-flat over the base ring. Section 14.3 contains some basic constructions of derived functors between categories of DG bimodules. Next, in Section 14.4, we define tilting DG bimodules. Among other results, we prove that a DG B-A-bimodule T is tilting iff T is a compact generator on the B side and it has the NC derived Morita property on the B side. We also prove the Rickard--Keller Theorem.

In Section 14.5 we introduce the NC derived Picard group of a ring A. The structure of this group is calculated when A is either local or commutative.